STRUCTURAL ANALYSIS – I Syllabus: - dnrcet.org

STRUCTURAL ANALYSIS – I

Syllabus:

UNIT – I

PROPPED CANTILEVERS: Analysis of propped cantilevers-shear force and Bending

moment diagrams-Deflection of propped cantilevers.

UNIT – II

FIXED BEAMS – Introduction to statically indeterminate beams with U. D. load central

point load, eccentric point load. Number of point loads, uniformly varying load, couple

and combination of loads shear force and Bending moment diagrams-Deflection of fixed

beams effect of sinking of support, effect of rotation of a support.

UNIT – III

CONTINUOUS BEAMS: Introduction-Clapeyron’s theorem of three moments-

Analysis of continuous beams with constant moment of inertia with one or both ends

fixed-continuous beams with overhang, continuous beams with different moment of

inertia for different spans-Effects of sinking of supports-shear force and Bending moment

diagrams.

UNIT-IV

SLOPE-DEFLECTION METHOD: Introduction, derivation of slope deflection

equation, application to continuous beams with and without settlement of supports.

UNIT – V

ENERGY THEOREMS: Introduction-Strain energy in linear elastic system, expression

of strain energy due to axial load, bending moment and shear forces - Castigliano’s first

theorem-Deflections of simple beams and pin jointed trusses.

UNIT – VI

MOVING LOADS and INFLUENCE LINES: Introduction maximum SF and BM at a

given section and absolute maximum S.F. and B.M due to single concentrated load U. D

load longer than the span, U. D load shorter than the span, two point loads with fixed

distance between them and several point loads-Equivalent uniformly distributed load-

Focal length.

INFLUENCE LINES: Definition of influence line for SF, Influence line for BM- load

position for maximum SF at a section-Load position for maximum BM at a sections,

ingle point load, U.D. load longer than the span, U.D. load shorter than the span-

Influence lines for forces in members of Pratt and Warren trusses.

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UNIT - VI

-STRAIN ENERGY-

Introduction: - Strain energy is as the energy which is stored within a material when work has

been done on the material. Here it is assumed that the material remains elastic

whilst work is done on it so that all the energy is recoverable and no permanent

deformation occurs due to yielding of the material,

Strain energy U = work done

Thus for a gradually applied load the work done in straining the material will be

given by the shaded area under the load-extension graph of Fig.

U = P δ

Work done by a gradually applied load.

The unshaded area above the line OB of Fig. 7.1 is called the complementary

energy, a quantity which is utilized in some advanced energy methods of solution

and is not considered within the terms of reference of this text.







UNIT-VI

MOVING LOADS AND INFLUENCE LINES

Definitions of influence line

An influence line is a diagram whose ordinates, which are plotted as a function of

distance along the span, give the value of an internal force, a reaction, or a

displacement at a particular point in a structure as a unit load move across the

structure.

An influence line is a curve the ordinate to which at any point equals the value of

some particular function due to unit load acting at that point.

An influence line represents the variation of either the reaction, shear, moment, or

deflection at a specific point in a member as a unit concentrated force moves over the

member.

In engineering, an influence line graphs the variation of a function (such as the shear felt in a

structure member) at a specific point on a beam or truss caused by a unit load placed at any point

along the structure. Some of the common functions studied with influence lines include reactions

(the forces that the structure’s supports must apply in order for the structure to remain static),

shear, moment, and deflection (Deformation). Influence lines are important in designing beams

and trusses used in bridges, crane rails, conveyor belts, floor girders, and other structures where

loads will move along their spanThe influence lines show where a load will create the maximum

effect for any of the functions studied.

Influence lines are both scalar and additiveThis means that they can be used even when the load

that will be applied is not a unit load or if there are multiple loads applied. To find the effect of

any non-unit load on a structure, the ordinate results obtained by the influence line are multiplied

by the magnitude of the actual load to be applied. The entire influence line can be scaled, or just

the maximum and minimum effects experienced along the line. The scaled maximum and

minimum are the critical magnitudes that must be designed for in the beam or truss

In cases where multiple loads may be in effect, the influence lines for the individual loads may

be added together in order to obtain the total effect felt by the structure at a given point. When

adding the influence lines together, it is necessary to include the appropriate offsets due to the

spacing of loads across the structure. For example, a truck load is applied to the structure. Rear

axle, B, is three feet behind front axle, A, then the effect of A at x feet along the structure must

be added to the effect of B at (x – 3) feet along the structure—not the effect of B at x feet along

the structure.

Many loads are distributed rather than concentrated. Influence lines can be used with either

concentrated or distributed loadings. For a concentrated (or point) load, a unit point load is

moved along the structure. For a distributed load of a given width, a unit-distributed load of the

same width is moved along the structure, noting that as the load nears the ends and moves off the

structure only part of the total load is carried by the structure. The effect of the distributed unit

load can also be obtained by integrating the point load’s influence line over the corresponding

length of the structures.

https://en.wikipedia.org/wiki/Beam_(structure)

https://en.wikipedia.org/wiki/Truss

https://en.wikipedia.org/wiki/Shear_force

https://en.wikipedia.org/wiki/Bending_moment

https://en.wikipedia.org/wiki/Deflection_(engineering)

https://en.wikipedia.org/wiki/Bridge

https://en.wikipedia.org/wiki/Conveyor_belt

https://en.wikipedia.org/wiki/Scalar_(mathematics)

https://en.wikipedia.org/wiki/Additive_map

1) A system of concentrated load, role beam left to right, s.s beam span of 10m and 10 KN

load leading

Find 1.Absolute max +ve S.F

2. .Absolute max -ve S.F

3..Absolute max BM

Solution

1. Absolute max +ve S.F

Using the similar triangle method and we get the x, y & z values

X = 0.85 m

Y = 0.75 m

Z = 0.55 m

S.F = (10×1)+(15×0.83)+(20×0.75)+(10×0.55)

= 43.25 KN

Using the similar triangle method and we get the l, m, n & o values

L=0.8 m

M = 0.65 m

N = 0.55 m

O = 0.35 m

S.F = (20×1)+(10×0.8)+(15×0.65)+(20×0.55)+(10×0.35)

= 52.25 KN

Absolute max -ve S.F


L = 0.35 m

M = 0.55 m

N = 0.7 m

O = 0.8 m

S.F = (10×1)+(20×0.8)+(15×0.7)+(10×0.55)+(20×0.35)

= - 49 KN

Using the similar triangle method and we get the l,m, & n values

L=0.55 m

M=0.75 m

N=0.85 m

S.F=-((20×1)+(15×0.9)+(10v0.75)+(20×0.55))=-52 KN

(iii) Absolute max BM


L = 0.75 m

M = 1.75 m

N = 2 m

O = 1 m

Max BM = (20×0.75) +(10×1.75)+(15×2.5)+(20×2)+(10×1)

= 22.75 KN

2) The four equal loads of 150 KN ,each equally spaced at apart 2m and UDL of 60 KN/m

at a distance of 1.5m from the last 150 KN loads cross a girder of 20m from span R to

L.Using influence line ,calculate the S.F and BM at a section of 8m from L.H.S

support when leading of 150KN 5m from L.H.S.

Solution

(i) Max BM

L = 3 m

M = 4.2 m

N = 4.4 m

0 = 3.6 m

P = 3 m

A = 11.25 m2

BM = (150×3)+(150×4.2)+(150×4.4)+(150×306)+(60×11.25)

= 2955 KNm

ii) Shear Force

Compute maximum end shear for the given beam loaded with moving loads as shown in

Figure

L = 0.25 m,

M = 0.3 m,

N = 0.55 m,

O = 0.45 m,

P = 0.375 m

SF = ((150×0.25)+(150×0.35)+(150×0.55)+(150×0.45)+(60×1.41))

= 144. KN

Where do you get rolling loads in practice?

Shifting of load positions is common enough in buildings. But they are more

pronounced in bridges and in gantry girders over which vehicles keep rolling.

Name the type of rolling loads for which the absolute maximum bending moment occurs at

the midspan of a beam.

Single concentrated load

udl longer than the span

udl shorter than the span

Also when the resultant of several concentrated loads crossing a span, coincides with

a concentrated load then also the maximum bending moment occurs at the centre of

the span.

What is meant by absolute maximum bending moment in a beam?

When a given load system moves from one end to the other end of a girder, depending

upon the position of the load, there will be a maximum bending moment for every

section.

The maximum of these bending moments will usually occur near or at the midspan.

The maximum of maximum bending moments is called the absolute maximum

bending moment.

Where do you have the absolute maximum bending moment in a simply supported beam

when a series of wheel loads cross it?

When a series of wheel loads crosses a simply supported beam, the absolute

maximum bending moment will occur near midspan under the load Wcr , nearest to

midspan (or the heaviest load).

If Wcr is placed to one side of midspan C, the resultant of the load system R shall be

on the other side of C; and Wcr and R shall be equidistant from C.

Now the absolute maximum bending moment will occur under Wcr .

If Wcr and R coincide, the absolute maximum bending moment will occur at

midspan.

What is the absolute maximum bending moment due to a moving udl longer than the span

of a simply supported beam?

When a simply supported beam is subjected to a moving udl longer than the span, the

absolute maximum bending moment occurs when the whole span is loaded.

Mmax max = wl2/ 8

State the location of maximum shear force in a simple beam with any kind of loading.

In a simple beam with any kind of load, the maximum positive shear force occurs at

the left hand support and maximum negative shear force occurs at right hand support.

What is meant by maximum shear force diagram?

Due to a given system of rolling loads the maximum shear force for every section of

the girder can be worked out by placing the loads in appropriate positions.

When these are plotted for all the sections of the girder, the diagram that we obtain is

the maximum shear force diagram.

This diagram yields the ‘design shear’ for each cross section.

What is meant by influence lines?

An influence line is a graph showing, for any given frame or truss, the variation of

any force or displacement quantity (such as shear force, bending moment, tension,

deflection) for all positions of a moving unit load as it crosses the structure from one

end to the other.

What are the uses of influence line diagrams?

Influence lines are very useful in the quick determination of reactions, shear force,

bending moment or similar functions at a given section under any given system of

moving loads and

Influence lines are useful in determining the load position to cause maximum value of

a given function in a structure on which load positions can vary.

Draw the influence line diagram for shear force at a point X in a simply supported beam

AB of span ‘l’ m.

Draw the ILD for bending moment at any section X of a simply supported beam and mark

the ordinates.

What do you understand by the term reversal of stresses?

In certain long trusses the web members can develop either tension or compression

depending upon the position of live loads.

This tendancy to change the nature of stresses is called reversal of stresses.

State Muller-Breslau principle.

Muller-Breslau principle states that, if we want to sketch the influence line for any

force quantity (like thrust, shear, reaction, support moment or bending moment) in a

structure,

We remove from the structure the resistant to that force quantity and

We apply on the remaining structure a unit displacement corresponding to that force

quantity.

The resulting displacements in the structure are the influence line ordinates sought.

State Maxwell-Betti’s theorem.

In a linearly elastic structure in static equilibrium acted upon by either of two systems

of external forces, the virtual work done by the first system of forces in undergoing

the displacements caused by the second system of forces is equal to the virtual work

done by the second system of forces in undergoing the displacements caused by the

first system of forces.

Maxwell Betti’s theorem helps us to draw influence lines for structures.

What is the necessity of model analysis?

When the mathematical analysis of problem is virtually impossible.

Mathematical analysis though possible is so complicated and time consuming that the

model analysis offers a short cut.

The importance of the problem is such that verification of mathematical analysis by

an actual test is essential.

Define similitude.

Similitude means similarity between two objects namely the model and the prototype

with regard to their physical characteristics:

Geometric similitude is similarity of form

Kinematic similitude is similarity of motion

Dynamic and/or mechanical similitude is similarity of masses and/or

forces.

State the principle on which indirect model analysis is based.

The indirect model analysis is based on the Muller Breslau principle.

Muller Breslau principle has lead to a simple method of using models of structures to

get the influence lines for force quantities like bending moments, support moments,

reactions, internal shears, thrusts, etc.,

To get the influence line for any force quantity,

(i) remove the resistant due to the force,

(ii) apply a unit displacement in the direction

(iii) plot the resulting displacement diagram.

This diagram is the influence line for the force.

What is the principle of dimensional similarity?

Dimensional similarity means geometric similarity of form.

This means that all homologous dimensions of prototype and model must be in some

constant ratio.

What is Begg’s deformeter?

Begg’s deformeter is a device to carry out indirect model analysis on structures.

It has the facility to apply displacement corresponding to moment, shear or thrust at

any desired point in the model.

In addition, it provides facility to measure accurately the consequent displacements all

over the model.

Name any four model making materials.

Perspex,

plexiglass,

acrylic,

plywood,

sheet araldite

bakelite

Micro-concrete,

mortar and plaster of paris

What is ‘dummy length’ in models tested with Begg’s deformeter.

Dummy length is the additional length (of about 10 to 12mm) left at the extremities of

the model to enable any desired connection to be made with the gauges.

What are the three types of connections possible with the model used with Begg’s

deformeter.

Hinged connection

Fixed connection

Floating connection

What is the use of a micrometer microscope in model analysis with Begg’s deformeter.

Micrometer microscope is an instrument used to measure the displacements of any

point in the x and y directions of a model during tests with Begg’s deformeter.

Construct the influence line for the reaction at support B for the beam of span 10 m. The

beam structure is shown in Figure

Solution:

A unit load is places at distance x from support A and the reaction value RB is

calculated by taking moment with reference to support A.

Let us say, if the load is placed at 2.5 m. from support A then the reaction RB can be

calculated as follows

Σ MA = 0:

RB x 10 - 1 x 2.5 = 0 ⇒ RB = 0.25

Similarly, the load can be placed at 5.0, 7.5 and 10 m away from support A and

reaction RB can be computed and ta bulated as given below.

X RB

0 0

2.5 0.25

5 0.5

7.5 0.75

10 1

Graphical representation of influence line for RB is shown in Figure

Influence Line Equation:

When the unit load is placed at any location between two supports from support A at

distance x then the equation for reaction RB can be written as

Σ MA = 0:

RB x 10 – x = 0 ⇒ RB = x/10

Find the maximum positive live shear at point C when the beam as shown in figure, is

loaded with a concentrated moving load of 10 kN and UDL of 5 kN/m.

Concentrated load:

the maximum live shear force at C

will be when the concentrated load

10 kN is located just before

C or just after C.

Our aim is to find positive live

shear and hence, we will put 10 kN

just after C.

In that case, Vc = 0.5 x 10 = 5 kN.

UDL:

the maximum positive live shear

force at C willbe when the

UDL 5 kN/m is acting between

x = 7.5 and x = 15.

Vc = [ 0.5 x (15 –7.5) (0.5)] x 5 = 9.375

Total maximum Shear at C:

(Vc) max = 5 + 9.375 = 14.375.

Muller Breslau Principle for Qualitative Influence Lines

In 1886, Heinrich Müller Breslau proposed a technique to draw influence lines

quickly.

The Müller Breslau Principle states that the ordinate value of an influence line for any

function on any structure is proportional to the ordinates of the deflected shape that is

obtained by removing the restraint corresponding to the function from the structure

and introducing a force that causes a unit displacement in the positive direction.

Procedure:

First of all remove the support corresponding to the reaction and apply a force in the

positive direction that will cause a unit displacement in the direction of RA

Deflected shape of beam

The resulting deflected shape will be proportional to the true influence line for the

support reaction at A.

Influence line for support reaction A

The deflected shape due to a unit displacement at A is shown in above Figure:1 and

matches with the actual influence line shape as shown in Figure 3.

Note that the deflected shape is linear, i.e., the beam rotates as a rigid body without

any curvature. This is true only for statically determinate systems.

Overhang beam


Now apply a force in the positive direction that will cause a unit displacement in the

direction of VC.

The resultant deflected shape is shown above Figure. Again, note that the deflected

shape is linear.

Influence line for shear at section C

Overhang beam - 2

Beam structure

To construct influence line for moment, we will introduce hinge at C and that will

only permit rotation at C.

Now apply moment in the positive direction that will cause a unit rotation in the

direction of Mc.

The deflected shape due to a unit rotation at C is shown in Figure and matches with

the actual shape of the influence line as shown in Figure 3.


Influence line for moment at section C

Maximum shear in beam supporting UDLs

UDL longer than the span

Influence line for moment at section C

Suppose the section C is at mid span, then maximum moment is given by

UDL longer than the span

Influence line for support reaction at A

Influence line for support reaction at B

Influence line for shear at section C

Maximum negative shear is given by

Maximum positive shear is given by

STRUCTURAL ANALYSIS – I Syllabus: - dnrcet.org

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